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Questions and Answers
What is the definition of the directional derivative of a function?
The directional derivative of a function f in the direction of a unit vector U is given by the limit as h approaches 0 of the difference quotient (f(x + h cos θ, y + h sin θ) - f(x, y)) / h.
What does DU f(x, y) denote?
DU f(x, y) denotes the directional derivative of the function f at the point (x, y) in the direction of the unit vector U.
The partial derivatives fx and fy are special cases of the directional derivative.
True
In the definition of the directional derivative, which of the following represents the function's rate of change?
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If U is the unit vector cos θi + sin θj, what is the limit as h approaches 0 used for?
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Study Notes
Directional Derivative
- The directional derivative of a function f(x, y) in the direction of a unit vector U (cos θi + sin θj) is given by:
- DU f(x, y) = lim (h→0) [f(x + h cos θ, y + h sin θ) - f(x, y)] / h
- The directional derivative represents the rate of change of the function values f(x, y) with respect to the direction of the unit vector U.
- Partial derivatives (fx and fy) are special cases of the directional derivative in the directions of i and j, respectively.
Example
- The directional derivative of f(x, y) = 3x^2 - y + 4x^2 in the direction of U = (cos(π/6), sin(π/6)) is given by:
- DU f(x, y) = lim (h→0) [f(x + (h * cos(π/6)), y + (h * sin(π/6))) - f(x, y)] / h
- Simplifying, we get: DU f(x, y) = lim (h→0) [3(x + (h * √3/2))^2 - (y + (h/2)) + 4(x + (h * √3/2))^2 - (3x^2 - y + 4x^2)] / h
- This expression can be further simplified and evaluated for specific values of x and y.
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Description
This quiz focuses on the concept of the directional derivative in multivariable calculus. It covers the definition, calculation, and practical examples of directional derivatives, emphasizing the role of unit vectors. Perfect for students looking to deepen their understanding of how functions change in specific directions.
